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Stochastic Analog Circuit Behavior Modeling by Point Estimation Method Fang Gong University of California, Los Angeles Electrical Engineering Department Los Angeles, CA 90095, US [email protected] Hao Yu Nanyang Technological University Electrical and Electronic Engineering [email protected] ABSTRACT Stochastic device parameter variations have dramatically increased beyond the scale of 65nm and can significantly lead to large mismatch for analog circuits. To estimate unknown analog circuit behavior in performance space under the given stochastic variations in parameter space, many state-of-art approaches have been developed recently. However, either Gaussian distribution or response surface model (RSM) with analytical formulae has to be assumed when connecting performance space and parameter space. A novel point-estimation based approach has been proposed in this paper to capture arbitrary stochastic distributions for analog circuit behaviors in performance space. First, to evaluate high-order moments of circuit behavior in an accurate fashion, the point-estimation method has been applied with only a few number of simulations. Then, probability density function (PDF) of circuit behavior can be efficiently extracted by the obtained high-order moments. This method is further extended for multiple parameters under linear complexity. Extensive numerical experiments on a number of different circuits have demonstrated that the proposed pointestimation method can provide up to 181X runtime speedup with the same accuracy, when compared with Monte Carlo method. Moreover, it can further achieve up to 15X speedup over the RSM-based method such as APEX with the similar accuracy. ACM Classification Keywords: B.7.2: - Integrated CircuitsDesign Aids General Terms: Algorithms, Performance Authors Keywords: Behavior Modeling, Point Estimation, Circuit simulation. 1. INTRODUCTION As semiconductor industry enters into nano-technology node, large process variations become inevitable and hence pose a serious threat to both analog circuit design and man- Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISPD’11, March 27–30, 2011, Santa Barbara, California, USA. Copyright 2011 ACM 978-1-4503-0550-1/11/03 ...$10.00. Lei He University of California, Los Angeles Electrical Engineering Department Los Angeles, CA 90095, US [email protected] ufacturing [1, 2, 3, 4]. Device variables in parameter space, such as the effective channel length and threshold voltage of transistors, can deviate significantly from nominal values due to large uncertainties from chemical mechanical polishing (CMP), etching, lithography and etc. Under such circumstance, circuit behaviors in performance space can differ from the nominal case by a large margin, which may further lead to high loss of yield. With the process variation of device variables in parameter space, it is desirable to extract the unknown distribution of variable circuit behaviors in performance space. A robust circuit design and yield enhancement are especially important for analog circuits. One critical but missing link here is how to find an efficient yet accurate mapping between parameter space and performance space. Note that the local random or stochastic variation is the most difficult one to be calculated, which is also called mismatch for the behavior modeling of analog circuits. In the past decade, many stochastic techniques had been proposed, such as Monte Carlo simulation, linear regression [1], stochastic orthogonal polynomials (SoPs) expansion [5, 3], response surface modeling based approaches [2, 4] and etc. The most general approach is to apply the Monte Carlo (MC) simulation, which samples all variable parameters and then calculate stochastic analog circuit behaviors by a large number of repeated simulations. As such, MC is too time-consuming to be afforded for the post-layout verifications beyond 65nm. To relieve the computational complexity, linear regression method [1] has been deployed to approximate the circuit behavior in performance space by a linear function of a number of normally distributed process variables in parameter space, or Gaussian distribution. This approach is efficient because of using analytical formula to obtain the circuit behavior. However, this approach cannot approximate nonnormal (non-Gaussian) distributions and might lead to the loss of accuracy. With the use of different stochastic orthogonal polynomials (SoPs), SoP based methods can model process variations with non-Normally distributed random variables. The unknown distribution of circuit behaviors in performance space can be estimated by solving SoP expansion coefficients [5, 3]. However, the SoP-based methods require knowing the type of the stochastic distribution of the circuit behavior. In practice, one known parameter distribution in parameter space usually becomes unknown in performance space after the mapping. In order to capture unknown random distribution after the mapping, response-surface-method (RSM) based methods [6, 4, 2] have been developed. One most important work developed recently is asymptotic probability extraction (APEX) [2] with the use of asymptotic waveform evaluation [7]. This approach assumes a polynomial function of all process parameters and further applies moment matching to extract the random distribution of circuit behavior (e.g. delay, gain, etc.). Nevertheless, the limitations of RSM-based approaches can be summarized in two-fold. First, the circuit behavior in performance space has become one strongly nonlinear function for random device variables. As such, the extraction by RSM has become computationally expensive. Second, it is prohibitive to evaluate high-order moments E(f k ) with analytical formula of f , especially when the number of random variables and the moment order k increase. As such, the approaches based on RSM still cannot mitigate the super-linearly complexity while remaining accuracy for large-scale problems. In this paper, a new mapping algorithm is developed to obtain the arbitrary circuit behavior in performance space from the arbitrary device variable in parameter space. Highorder moments of circuit behavior f are first estimated by Point Estimation (PE) method to efficiently characterize the high-order moments E(f k ) by weighted-sum of a few sampled simulations. Therefore, one can significantly improve the efficiency and accuracy of APEX [2] without the need to assume RSM inputs. As a result, the distribution of circuit behavior f in performance space can be efficiently obtained by its moments E(f k ), calculated from the PE method in the parameter space. Moreover, a normalized PDF function is introduced so that to enhance the accuracy by eliminating the potential round-off error. In addition, this approach has been extended to consider the case with multiple parameters. Extensive experiments on a number of different circuits are performed to demonstrate the validity and efficiency of our proposed algorithm. The contributions of this paper are further clarified as follows. First, although point estimation method has widely been applied for reliability analysis [8, 9], it can only estimate at most four moments (e.g. the mean, the variance, the skewness, the kurtosis) with empirical analytical formulae, and hence remains unclear how to estimate moments with higher order. In this paper, a modified point estimation method is developed to approximate higher order moments in a systematic manner. Moreover, unlike the observed super-linearly complexity in RSM based methods, our proposed method can be extended to deal with multiple parameters with linear complexity, which is significant for large-scale analog circuits. The rest of this paper is organized as follows. In Section 2, we first review the mathematical formulation of the PDF estimation and the moments used in response-surface-model (RSM) methods. In Section 3, we introduce the point estimation (PE) method and further propose a new high-order moments evaluation via PE. We also discuss one normalized PDF technique in Section 4 to reduce error and further present experimental results in Section 5. This paper concludes in Section 6. 2. BACKGROUND 2.1 Mathematical Formulation We consider circuit behavior f with multiple random vari- ables of process variations (x1 , x2 , · · · , xn ), which can be expressed as f (x1 , x2 , · · · , xn ). As such, parameter space can be defined as the space Rn bounded by the min and max of all random variables, and performance space R consists of all possible behavior merits. As a result of uncertainties in process technology, random variables can deviate from their nominal values and lead to variational circuit behavior. Our purpose is to extract unknown distribution (e.g. PDF/CDF functions) of circuit behavior by mapping the variable parameter distributions in parameter space into performance space. To this end, the probabilistic moments in both spaces should be defined according to probability theory [10, 11]: mpf = E(f p ) = mpx = E(xp ) = +∞ R −∞ +∞ R (f p · pdf (f ))df (xp · pdf (x))dx (1) −∞ where mpf is the p-th moment of circuit behavior f in performance space, and mpx is the p-th moment of random variable x in parameter space. Lemma 1. Suppose pdf (f ) is continuous in performance space. Then pdf (f ) can be determined uniquely by high order moments E(f k ) (k = 1, 2, · · · , m). Proof. Let Φ(ω) is the Fourier transform [12] of pdf (f ) and can be written as: Φ(ω) = +∞ Z ³ ´ pdf (f ) · e−jωf df (2) −∞ = ! +∞à Z +∞ X (−jωf )p df pdf (f ) · p! p=0 −∞ = +∞ Z +∞ X (−jω)p · (f p · pdf (f ))df. p! p=0 −∞ = +∞ X p=0 p (−jω) · mpf . p! As such, Φ(ω) can be expanded with high order moments mpf , and pdf (f ) can be extracted from Inversion Fourier transform of Φ(ω). In other words, there is a one-to-one correspondence between high order moments mpf and pdf (f ). Notice that mpx in (1) can be computed accurately in parameter space with known pdf (x). Therefore, it is the key problem to find an efficient mapping between mpx and mpf in order to extract pdf (f ) in performance space. 2.2 Preliminary of PDF Calculation The techniques to extract pdf (f ) with moments E(f k ) have been proposed in [2, 7], which will be reviewed in what follows. First, time moments for f can be defined as: (−1)k · mf = k! _k +∞ Z f k · pdf (f )df . (3) −∞ _k It is clear that mf is defined in performance space ±but different from mkf in (1) due to a scaling factor (−1)k k!. On the other hand, consider a linear time-invariant (LTI) system H, and its time moments can also be defined as[7]: _k mt = (−1)k · k! +∞ Z tk · h(t)dt. (4) −∞ where t is the time variable and h(t) is impulse response of LTI system H. So, impulse response h(t) can be an optimal approximation to pdf (f ) if we treat t as circuit behavior f _k _k and make mt equal to mf . Furthermore, time moments in (4) can be expressed as [7]: _k mt = − M X ar . k+1 b r=1 r (5) Where ar and br (r = 1, · · · , M ) are the residues and poles of this LTI system, respectively. As such, the impulse response of the LTI system can be simplified as: M k+1 P ar · ebr ·t (t ≥ 0) h(t) = (6) r=1 0 (t < 0) In general, there are three steps to calculate h(t) as an approximation to pdf (f ): • Mapping mkx in parameter space into performance space _k as mf in (3). _k _k • Make mf equal to mt and solve nonlinear equation system in (5) for residues ar and poles br . • Compute impulse response h(t) in (6) with residues ar and poles br . Clearly, one needs to find an efficient mapping between parameter space and performance space to obtain the stochastic circuit behavior. Within this mapping, the most challenging step is the evaluation of high-order moments. Although the RSM based methods, such as APEX, assume that one nonlinear function can be found to approximate the mapping, they might become unaffordable to calculate the high-order moments for large-scale stochastic problems. To this end, we have developed a modified point estimation (PE) method to perform the mapping for the calculation of high-order moments. 3. HIGH ORDER MOMENTS ESTIMATION In this section, we discuss how to evaluate high order moments of circuit behavior mkf by mapping parameter moments mkx from parameter space into performance space via Point Estimation (PE) method. 3.1 Moments via Point Estimation For illustration purpose, we consider circuit behavior f (x) with single variable parameter x. Usually it is impractical to compute mkf as (1) because pdf (f ) is unknown. The other straightforward way is to use Taylor expansion, which involves high order derivatives. But there is no way to guarantee the existence of high order derivatives of f (x). As such, we propose to leverage the Point Estimation method to compute high order moments[8, 9], which approximates mkf with a weighted sum of sampling values of f (x). Assume x̃j (j = 1, · · · , p) are estimating points of random variable, and Pj are corresponding weights. In this way, the k-th order moment of f (x) can be approximated as: mkf +∞ Z p X = f k · pdf (f )df ≈ Pj · f (x̃j )k . (7) j=1 −∞ However, [8, 9] only provide empirical analytical formulae of x̃j and Pj for first four moments. Therefore, it is significant but remains unknown how to determine x̃j and Pj for higher order moments systematically. 3.2 Estimating Points and Weights To this end, we start with (7) in performance space, but it is impossible to compute x̃j and Pj since both sides are unknown. Thus, we need to reformulate the problem in parameter space, where random variable x and its distribution (e.g. PDF function pdf (x)) are known beforehand. According to classic probability theory[10, 11], we have following theorem: Theorem 1. Let x and f (x) are both continuous random variables, and their PDFs are pdf (x) and pdf (f ), respecR tively. Suppose f k (x) · pdf (x)dx exists. Then Z Z E(f k (x)) = f k (x) · pdf (f )df = f k (x) · pdf (x)dx As such, the moments of circuit behavior f (x) in performance space can be calculated in parameter space. For example, the k-th order moment of f (x) in equation (7) becomes: Z m X Pj · f (x̃j )k . (8) mkf = f (x)k · pdf (x)dx ≈ j=1 On the other hand, the k-th order moments of random variable x can be written as: Z m X 0 0 mkx = xk · pdf (x)dx ≈ Pj · (x̃j )k . (9) j=1 It is obvious that estimating points x̃j and corresponding 0 0 weights Pj in (8) are the same as x̃j and Pj in (9) because they are all defined in parameter space. Therefore, we can 0 0 calculate x̃j and Pj from equation (9) with mkx obtained from (1), and then estimate mkf using (8). 0 0 Now, the problem is how to solve for x̃j and Pj systematically. Since there are total 2m unknowns in (9), we need to build 2m equations using first 2m moments of random variable, which can be rewritten as: m P j=1 m P j=1 m P j=1 ··· m P j=1 0 Pj = 1 = m0x 0 0 Pj · x̃j = E(x) = m1x 0 0 0 0 Pj · (x̃j )2 = E(x2 ) = m2x (10) Pj · (x̃j )2m−1 = E(x2m−1 ) = m2m−1 x Note that the right-hand-side of above nonlinear system are first 2m moments of x in the behavior domain and can be calculated exactly with known pdf (x) and definition in (1). This nonlinear system (10) can be solved using algorithm proposed in [7]. In what follows, we briefly describe this algorithm. 0 0 Assume residues aj = Pj and poles bj = 1/x̃j , the equations (10) can be reformulated as: a1 + a2 + · · · am m0x a1 a2 am 1 + + · · · b1 b2 bm mx a1 2 + ab22 + · · · ab2m m b2 = x m 1 2 (11) . . . .. . a1 a2 am + 2m−1 + · · · 2m−1 m2m−1 x b2m−1 b b of single variable xi with other variables set equal to mean values. However, [8, 9] can only estimate first four moments using explicit analytical formulae as linear combination. Consider circuit behavior f (x1 , x2 , · · · , xn ) where x1 , x2 , · · · , xn are independent random variables, it is desirable to estimate mkf (x1 ,x2 ,··· ,xn ) that is the k-th order moment of f (x1 , x2 , · · · , xn ). To estimate higher order moments of f (x1 , x2 , · · · , xn ) systematically, we derive the equation for mkf (x1 ,x2 ,··· ,xn ) as: The system matrix of (11) is the well-known Vandermonde matrix and can be divided into two parts: Moreover, gi can be calculated as follows and detailed derivation can be referred to technical report: 1 mkf (x1 ,x2 ,··· ,xn ) = M · v = rhslow ; M · Λ−q · v = rhsupper where rhslow consists of the low order moments (k = 0,1, · · · , m − 1), and rhsupper contains the high order moments (k = m, m + 1, · · · , 2m − 1). Λ−1 is a diagonal matrix of {1/bj } (j = 1, · · · , m). And M matrix (m × m) can be expressed as: 1 1 ··· 1 −1 −1 −1 b1 b2 ··· bm−1 −2 −2 b b · · · b−2 1 2 m−1 M = .. .. .. .. . . . . −(m−1) b1 −(m−1) b2 ··· Therefore, the linear system M · v = rhslow can be solved as v = M −1 · rhslow , and rhsupper = M · Λ−q · M −1 · rhslow . Since M is also a Vandermonde matrix that is the modal matrix for a system matrix in companion form, rhsupper can be simplified as rhsupper = M̂ −q · rhslow , where M̂ −1 is 0 1 0 ··· 0 0 0 1 ··· 0 M̂ −1 = . (12) . . . . .. .. .. .. .. . s0 s1 s2 · · · sm−1 In this way, the eigenvalues of M̂ −1 are {1/bj } in (11). To calculate M̂ −1 , we need to compute {st } (t = 0, · · · , m − 1) with following equation system: m mx m0x m1x · · · mm−1 s0 x m+1 2 m m1x mx · · · mx s1 mx .. = .. . .. .. .. .. . . . . . . sm−1 m2m−1 mm−1 mm · · · m2m−2 x x x x (13) When {1/bj } are available, the {aj } can be calculated from 0 Equation (11). Therefore, the weights {Pj } and estimating 0 points {x̃j } can be computed systematically and used to compute mkf in equation (8). 3.3 gi = c · Extension to multiple parameters It is usually necessary to handle multiple variable parameters simultaneously in real-world problems. Thus, we discuss how to extend aforementioned techniques to deal with multiple parameters. Existing methods [8, 9] model mkf (x1 ,x2 ,··· ,xn ) as a linear combination of moments mkf (xi ) , where f (xi ) is a function (14) ∂ (f (xi )) ∂xi For each parameter xi , we have its estimating points x̃j and corresponding f (x̃j ). Hence, it is possible to calculate ∂ (f (xi ))/∂xi with finite difference method numerically. Besides, the constant c can be computed with: m0f (x1 ,x2 ,··· ,xn ) = n X gi m0f (xi ) = i=1 n X gi (15) i=1 , n ∂ f (x ) ∂ (f (xi )) P 1 ( i ). = c· =1⇒c= ∂xi ∂x i i=1 i=1 n X −(m−1) bm−1 gi mkf (xi ) . i=1 m 2 n X As such, high order moments of multi-variable function mkf (x1 ,x2 ,··· ,xn ) can be evaluated with moments of univariate function mkf (xi ) efficiently. Extensive experiments can demonstrate its validity and efficiency. 3.4 Error Estimation Theoretical maximum approximation error of point estimation method is analyzed in [13]. For the univariate case, the maximum approximation error to exact integral value in equation (1) can be governed by: ¯ ¯ +∞ ¯X ¯ Z ¯m ¯ k k ¯ ¯ ≤ α · k1/m . (16) P · f (x̃ ) − f (x) · pdf (f )df j j ¯ ¯ ¯ j=1 ¯ −∞ where α is a constant, and k is the order of moments. m is the number of estimating points x̃j for order k. As such, it implies that more estimating points for each variable should be used to reduce the estimation error of higher order moments. 4. PDF CALCULATION WITH MOMENTS 4.1 PDF/CDF Estimation with Moments With high order moments mkf available, the next step is to compute residues {ar } as well as poles {br } in (5) and impulse response h(t) [2, 7] in (6). To do so, we calculate _k _k mf in (3) and make them equal to mt in (5). The nonlinear equation system becomes: − a1 b1 a1 b2 1 a1 b3 1 a1 b2M 1 + + + aM bM aM b2 M aM b3 M a2 b2 a2 b2 2 a2 b3 2 + ··· + ··· + ··· .. . a2 + b2M + · · · 2 aM b2M M PDF on all discrete points as: _0 mf _1 mf _2 mf M P . = . . . pdfnorm (fp ) = (17) M P _M mf Normalized PDF for Error Prevention From (17), it is obvious that the accuracy of PDF approximation mainly depends on the accuracy of residues ar , poles br and moments estimation. However, there are roundoff error within moment estimation and the PDF approximation, which can lead to instability issue. In order to prevent potential error, we propose to normalize PDF calculated from equation (6) to cancel out the potential roundoff error. For illustration purpose, we take the roundoff error in moments as an example, and other roundoff error can be elim_k inated with the same way. Assume mf is the exact value of k-th time moment in equation (3), and m̃kf is the estimated _k value of k-th time moment. Also, we assume m̃kf = const·mf due to roundoff error, where const is a scaling constant. As such, when we use the direct solution in section 2, the scaling constant in both system matrix and right-handside vector of (13) can be canceled out. Hence, st (t = 0, · · · , m − 1) and thus eigenvalues of equation (12) (that is poles {1/bj }) are both exact values. Next, the nonlinear equation system (17) becomes: _0 a1 M + ab22 + · · · abM mf b1 _1 a21 + a22 + · · · a2M mf b1 b2 bM _2 a1 + ab32 + · · · ab3M b3 (18) − = const · mf 1 2 M .. .. . . a1 2 M _M + ba2M + · · · ba2M b2M m 1 2 f M Which leads to ãj = const · aj , where aj are exact values of residues. Therefore, PDF of f (x) is approximated with: pdf (f ) = M X _k+1 ãr · e b r ·f r=1 = M X _k+1 const · ar · e b r ·f r=1 M X _k+1 const · ar · e b r ·fp = ·fp _k+1 const · ar · e b r (21) ·fp (20) r=1 To normalize it, we can divide it with the total value of _k+1 ar · e b r r=1 N P M P ·fp _k+1 ar · e b r ·fp p=1 r=1 In this way, the scaling constant can be eliminated from the approximation of PDF and thus normalization improves the numerical stability of proposed algorithm. 4.3 Error Estimation Since the Fourier transform in equation (3) is unique, it is equivalent to evaluate the error of Φ(ω) in order to investigate the accuracy of PDF approximation with qth order moments. It is ideally to compare the difference between Fourier transform of estimated PDF and that of exact PDF. However, the exact PDF is usually not available. Instead, we use approximation with n+1 order moments as the exact value, and proceed to estimate the error. ¯ ¯ q+1 ¯ Φ (ω) − Φq (ω) ¯ ¯ (22) Error = ¯¯ ¯ Φq+1 (ω) ¯ q+1 ¯ q ¯ P (−jω)p ¯ P (−jω)p ¯ · mpf − · mpf ¯¯ p! p! ¯ p=0 ¯ p=0 ¯ = ¯ ¯ q+1 ¯ ¯ P (−jω)p p ¯ ¯ · m f p! ¯ ¯ p=0 ¯ ¯ ! à −1 ¯ ¯ q+1 p X ¯ ¯ (−jω)q+1 mpf (−jω) ¯. = ¯¯ · · q+1 ¯ (q + 1)! p! m ¯ ¯ f p=0 When |mpf | ≥ |mq+1 | ( p ≤ q + 1), above error estimation f can become: ¯ Ãq+1 !−1 ¯¯ ¯ X (−jω)p ¯ ¯ (−jω)q+1 ¯. Error ≤ ¯¯ · (23) ¯ p! ¯ (q + 1)! ¯ p=0 The same error estimation can be obtained for |mpf | ≤ |mq+1 | f by taking reciprocal of circuit behavior to shift the behavioral distribution. As such, the error estimation can be used to measure the accuracy of the approximation with first q-th order moments. When the approximation order increases, the error estimation should move to higher order as required. 4.4 (19) In order to eliminate the scaling constant, we propose to normalize PDF of f (x) as follows: First, we discretize f (x) into discrete points {fp (x)} (p = 1, · · · , N ). As such, PDF on p-th discrete point {fp (x)} can be expressed as: pdf (fp ) = r=1 N P M P p=1 r=1 which has a Vandermonde matrix similar to (11) and can be solved with the same technique in [7]. With poles {br } and residues {ar } available, PDF function can be approximated with equation (6). Notice that impulse response h(t) is zero for t < 0, but the PDF in real-life problems can be nonzero for f ≤ 0. In this case, PDF function can be shifted as [2] which can be demonstrated with our experiments. 4.2 _k+1 const · ar · e b r Complexity Analysis The Monte Carlo simulation requires to generate massive samples to cover the entire parameter space evenly, so number of simulations is pn where p is the number of samplings for every single variable and n is the total number of random variables. As for RSM based methods, we take APEX as an example where RSM formulation is the most time-consuming part. For example, when RSM uses n random variables and k order polynomial function, it has total Cn1 + Cn2 + · · · + Cnk terms and thus APEX requires Cn1 +Cn2 +· · ·+Cnk simulation samples. In other words, the complexity of APEX is O(nk ). When proposed algorithm handles total n variables and needs m estimating points for each variable, the total complexity is (m − 1) ∗ n + 1 (usually m ¿ n) or O(n). m − 1 denotes that estimating points of each variable includes the nominal point and nominal circuit simulation can be shared by all variables. Therefore, it has linear complexity. 0.9 5. 0.4 5.1 0.7 0.6 0.5 EXPERIMENTAL RESULTS We have implemented proposed algorithm in the MATLAB environment, and all experiments are carried out on a Linux server with a 2.4GHz Xeon processor and 4GB memory. We use a six-transistor SRAM cell and a two-stage operational amplifier to compare the accuracy and efficiency of proposed algorithm with APEX [2] and Monte Carlo simulation. As an illustration, we consider the threshold voltages of MOSFETs as independent random variables subject to process variations, but our algorithm can also handle other variation sources. SRAM Cell We first consider a typical design of 6T SRAM cell in Fig.(1) and investigate the access time failure of the SRAM cell during reading operation, which is determined by the voltage difference between BL B and BL. Monte Carlo with variations Nominal analysis 0.8 Tmax 0.3 0.2 0.1 0 0 2 4 6 8 10 12 Figure 2: BL B discharge behavior with Vth variations to deal with rare event in an SRAM. Therefore, we focus on reducing average error of the performance distribution in the entire range. We start from univariate case to validate proposed algorithm and then extend it to multi-variable case. We have implemented three other methods for comparison: • Monte Carlo simulation (MC): This is direct Monte Carlo simulation. WL=1 Vdd • APEX: Implementation of asymptotic probability extraction algorithm proposed in [2]. +5V • Point Estimation Method (PEM): Proposed algorithm that leverages the point estimation method. Mn2 BL_B=1 Mp6 Mp5 Q =1 Q_B=0 Mn4 5.1.1 Univariate Case BL=1 Mn3 Mn1 GND First, we consider one random variable (e.g. threshold voltage variation on M n2 ) to compare the accuracy of APEX and PEM against MC. The random distributions (PDF function) from these methods are plotted in Fig.(3). Note that the histogram from Monte Carlo simulation has been normalized to eliminate the effect of total number of samplings. Figure 1: Schematic of SRAM 6-T Cell 0.16 Initially, both BL B and BL are pre-charged to V dd, while Q B stores zero and Q stores one. When reading the SRAM cell, BL B starts to discharge from V dd and produces a voltage difference ∆V between it and BL. The time it takes BL B to produce a large enough voltage difference is called access time. Since process variations are inevitable, the access time of manufactured SRAM cells can deviate from nominal value. When access time is larger than acceptable maximum value Tmax , this leads to an access time failure. In our experiment, we consider threshold voltages of all MOSFETs as independent variables which are normally distributed with 30% perturbation from nominal values. As such, there are perturbations to the nominal discharge trajectory on BL B as shown in Fig.(2). Therefore, we can investigate the access time failure by capturing the random distribution of voltage on BL B at Tmax time-step: when voltage of BL B at Tmax is larger than its nominal value, the access time failure happens. Note that both PEM and APEX can not capture high precision in the tail region of CDF/PDF, which is required 0.14 PEM APEX 0.12 0.1 0.08 0.06 0.04 0.02 0 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 Figure 3: random distributions of BL B voltage at Tmax (Monte Carlo result has been normalized) When compared with Monte Carlo results, PEM can provide better accuracy than APEX, especially in the peak and right tail regions. However, APEX has better efficiency even if the same order of moments are used: APEX needs only 8.59 second with quadratic function in response surface model, while PEM requires 17.71 seconds with seven estimating points for one variable. It is because APEX use analytical formula which is suitable for low dimensions. It should be noticed that the number of required simulation samples in APEX will increases exponentially when more variables or a strongly nonlinear RSM required. 5.1.2 Multiple Variable Parameters Next, we consider all threshold voltages of six transistors are independent random variables, which are normally distributed with 30% perturbation from nominal values. Similarly, we attempt to compare all three methods, but it is prohibitive to implement APEX as response surface model becomes very complicated, especially when high order response surface model is required. Instead of original APEX, we calculate high order moments numerically using results from Monte Carlo simulations and extract PDF with technique in APEX, which is denoted as MMC+APEX. The random distributions from all methods are compared in Fig.(4). 10 independent variables are used to model random variation for each transistor [14]. As such, RSM using quadratic function has 1830 coefficients and thus APEX requires 1830 simulation samples as discussed in Section 4.4. However, PEM only needs 121 simulation samples when 3 estimating points are used for each independent variable, and achieves 15X speedup over APEX. 5.2 Operational Amplifier We further consider a two-stage operational amplifier in Fig. (5) and a negative feedback circuit in Fig.(6). We use this example to show that proposed method can estimate random distributions where circuit behavior is negative. Vdd +5V Mp5 Mp8 Mp7 Output Mp1 Input- Mp2 Input+ Is Mn6 Mn4 Mn3 Vss 0.14 -5V Figure 5: Schematic of Operational Amplifier 0.12 PEM MMC+APEX 0.1 Similar to SRAM cell example, we consider threshold voltages of all MOSFETs as independent variables which are normally distributed with 30% perturbation from nominal values. Note that the threshold voltages of input transistor pair (M p1 , M p2 ) should be kept the same to ensure the convergence of nonlinear system solver in circuit simulators. 0.08 0.06 0.04 R2 0.02 R1 0 0.08 0.1 0.12 0.14 0.16 0.18 Output 0.2 Input Figure 4: random distributions of BL B voltage at Tmax (Monte Carlo result has been normalized) It is obvious that PEM can capture the exact distribution of BL B voltage at Tmax , which fits well with Monte Carlo simulation and MMC+APEX method. This can shows that PEM can achieve very high accuracy in the multi-variable problems. Also, we compare only the runtime of MC and PEM in Table 1, since runtime of MMC+APEX is almost the same as MC. In this table, PEM uses 5 estimating points for each variable parameter, and can achieve the same accuracy with 119.2X speedup over Monte Carlo method. Table 1: Runtime Comparison of three methods Method 3 Monte Carlo (3 × 10 ) PEM (5 point) Time (second) Speedup 7644 64.12 1x 119.2x To compare the efficiency with APEX, we can consider a SRAM cell under commercial 65nm CMOS process where Figure 6: Schematic of a unity gain feedback circuit There are a number of op am specifications in time-domain and frequency-domain, such as slew rate, settling time, phase margin, input offset voltage and etc. In our experiment, we investigate the input offset voltage variation due to threshold voltage variations of MOSFETs. In this experiment, we implement following methods for comparison purpose: • Monte Carlo simulation (MC): This is direct Monte Carlo simulation. • Moments from Monte Carlo (MMC+APEX): Estimate high order moments of input offset voltage from Monte Carlo simulation, and extract the PDF using techniques in [2, 7]. • Point Estimation Method (PEM): Proposed algorithm that leverages the point estimation method. First, we validate the accuracy of PEM by comparing the random distributions of input offset voltage from different methods in Fig.(7). PEM employs 5 estimating points for each variable and achieve the correct PDF function by shifting distribution of circuit behavior into positive region and moving it back. 0.35 MMC+APEX PEM 0.3 7. REFERENCES 0.25 0.2 0.15 0.1 0.05 0 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Figure 7: random distributions of input offset voltage (Monte Carlo result has been normalized.) Since MMC+APEX method has the exact moment values from Monte Carlo simulations, it can provide very high accuracy. Also, PEM can offer the same accuracy with MMC+APEX method, which implies that PEM can achieve high accuracy of high order moments. On the other hand, we further validate accuracy of PDF function from PEM against the histogram from Monte Carlo simulation in Fig.(7), which fit with each other very well. Moreover, we compare the runtime of Monte Carlo method and PEM in Table (2), and MMC+APEX is omitted because it has almost the same runtime as Monte Carlo method. Besides, we list PEM method with different number of estimating points for each variable to demonstrate it has linear scalability with the estimating points at the same time. Table 2: Runtime Comparison between Monte Carlo simulation and PEM Method 3 Monte Carlo (3 × 10 ) PEM (3 point) PEM (9 point) Time (second) Speedup 27765 (7.71 hours) 153.1 (0.04 hours) 525.84 (0.15 hours) 1x 181.5x 52.8x From Table (2), PEM can provide up to hundreds times speedup over MC method. Moreover, the computational cost increases linearly with the number of estimating points for each variable. 6. ically reduce the computational cost seen in APEX. Moreover, the proposed method can be extended to deal with multiple parameters under linear complexity. Experiments on a few different circuits have shown that the proposed method can provide up to 181X more runtime speedup with the same accuracy when compared with the Monte Carlo method. Also, it can achieve up to 15X speedup over the RSM based method such as APEX with the similar accuracy. 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